Formalizing dimension and response violations of local independence in the unidimensional Rasch model

Local independence in the Rasch model can be violated in two generic ways that are generally not distinguished clearly in the literature. In this paper we distinguish between a violation of unidimensionality, which we call trait dependence, and a specific violation of statistical independence, which we call response dependence, both of which violate local independence. Distinct algebraic formulations for trait and response dependence are developed as violations of the dichotomous Rasch model, data are simulated with varying degrees of dependence according to these formulations, and then analysed according to the Rasch model assuming no violations. Relative to the case of no violation it is shown that trait and response dependence result in opposite effects on the unit of scale as manifested in the range and standard deviation of the scale and the standard deviation of person locations. In the case of trait dependence the scale is reduced; in the case of response dependence it is increased. Again, relative to the case of no violation, the two violations also have opposite effects on the person separation index (analogous to Cronbach's alpha reliability index of traditional test theory in value and construction): it decreases for data with trait dependence; it increases for data with response dependence. A standard way of accounting for dependence is to combine the dependent items into a higher-order polytomous item. This typically results in a decreased person separation index index and Cronbach's alpha, compared with analysing items as discrete, independent items. This occurs irrespective of the kind of dependence in the data, and so further contributes to the two violations not being distinguished clearly. In an attempt to begin to distinguish between them statistically this paper articulates the opposite effects of these two violations in the dichotomous Rasch model.